Transcript Lecture 14 hydrodynamics lecture

Hydrodynamics
•How organisms cope with the forces imposed on
them by a dense and viscous medium
The Boundary layer
At the interface between moving water and a stationary substrate,
the water velocity is 0, i.e. “no slip” condition
This means that there is a sharp shear or gradient in velocity near
the substrate
It is within this velocity gradient that viscosity exerts its friction
We call the gradient region a boundary layer
U
U
y
Boundary layer
thickness 
This mayfly minimizes the force that the flowing water exerts on its
body by:
•Its flattened shape allowing it to occupy the boundary layer
•Its behaviour which allows it to graze attached algae without breaking
contact with the rock surface
Feeding at
the boundary
layer—black fly
larvae
Diptera
Simuliidae
Aquatic organisms experience a drag force from their hydrodynamic
environment
Drag force
CDrU2A
The fish swimming toward the left experiences a drag force
acting in the opposite direction
Actively moving animals like fishes don’t avoid drag like
mayflies do, but actually swim against this force at
considerable energetic cost. Their streamline shape however
helps minimize this cost
The Drag force results from a complex of hydrodynamic phenomena
(a) Inertial drag is the momentum flux pushing against
the anterior surface of the fish = rU2A
U
(b)Viscous drag results from
Skin friction
l
friction between
the sheets of fluid in the
boundary layer
=mUA/l
(c) Pressure drag results from separation of the
boundary flow from the body surface
producing a wake with turbulent eddies,
As a result pressure at the rear << pressure at the front
The ratio of the inertial force (dynamic pressure) over the viscous
force produces a dimensionless constant called the Reynolds number
rU 2 A lU
m
R

, where v  or the kinematic viscosity
mUA
v
r
l
Remember that the formula for drag = CDrU2A
The drag coefficient (CD)is not a constant, but rather changes
greatly with R. Normally it decreases sharply with R, as
viscous forces (mainly skin friction) become much less
important than inertial forces
CD
R
The Reynolds number (R) is a very useful indicator of the
hydrodynamic regime
It can range over many orders of magnitude: For example,
A large whale swimming at 10 m/sec
R=3 x 108
A tuna swimming at the same speed
R=3 x 107
A small trout going 1m/sec
R=3 x 105
A large copepod moving 10 cm/sec
R=200
A Daphnia moving 1 cm/sec
R=50
A rotifer moving 0.1cm/sec
R=2
An 10 um diatom sinking at 1m/day
R=0.001
Why aren’t small zooplankton streamlined in shape like fish?
How much big a burst of energy will a pike expend to capture prey?
Force required = drag, CDrU2A
Assume the pike is 50 cm long, and has a surface Area of 0.01m2
When striking prey such a fish can reach a speed of 2 m/sec
The density of water is around 1000 kg/m3
The R for such a fish is around 1 x 10 6, which means the wake
will be turbulent, and the CD will be around 0.02
Assume that the burst takes 1 sec, and covers 2m
Ul
0.5m  2m / s
6
R


1

10
v 1.00  10 6 m 2 / s
m 1.00  10 3 kg / m / s
6
2
v 

1

10
m
/s
3
3
r
1.00  10 kg / m
How much big a burst of energy will a pike expend to capture prey?
Force required = drag, CDrU2A
Force = (0.02)(1000 kg/m3)(2m/s)2(0.01m2) = 0.8 Newtons
Energy/Work = Force x distance,
Energy expenditure/s = 0.8 N x 2 m/s = 1.6 Joules / s
1J/s=1W, which is a power unit
In Power units, this burst requires 1.6 W to overcome the drag force
How profitable is this ?
•A 1 g perch contains around 6 kJ of energy and the fish can
assimilate at least half of this. So clearly to invest 1.6 J to obtain
3000 J is highly profitable
•On the other hand a small freshwater shrimp (0.01 g) would only
contain 10 J, of which the fish would likely assimilate less than half.
•Large fish generally won’t expend large bursts of energy to get small
prey because the profit margin isn’t as large and they can catch them
without spending a lot of energy.
How profitable is this ?
•If however the pike caught the smaller prey while swimming
casually at 0.5 m/sec, keeping its CD down to 0.007, the swimming
expenditure would only be (0.007)(1000)(0.5)(0.5)(0.01)(0.5)/s
=0.01 J/sec
•This energy output can be easily supplied aerobically by basal
metabolism.
Another case where hydrodynamics plays an important role
in aquatic ecology is in regard to the sedimentation of small particles
Stokes law gives the viscous drag force on a small spherical particle
in terms of U, m , and the linear dimension (r )
D  6mUr
The terminal velocity of the sinking particle will be the value of
U where the weight differenti al between th e particle
and an equivalent volume of water balances the drag force
( rp  rw)( 43 r 3 ) g  6mUr
2r 2 g ( rp  rw)
Ut 
9m
This equation w ould be expected to work for R  1
Clearly to minimize sinking speed phytoplank ton need to
keep r and ( rp  rw) low
Departure from spherical shape will also increase viscous drag
because it will add to r on the RHS, without increasing volume
Increasing the linear dimension without increasing volume will
increase D, the viscous drag because it will provide more surface
for friction t o act
D  6mUr
But if volume is not increased as well the LHS won' t change
( rp  rw)(V ) g
Since phytoplank ton benefit by minimizing sinking speed
(stay in the photic zone longer) we should expect to see
phytoplank ton, especially the larger species, depart considerab ly
from spherical shape.
Shapes of phytoplankton